code: def phillips_ouliaris_pval(
      stat: float,
      test_type: Literal["Za", "Zt", "Pu", "Pz"],
      trend: UnitRootTrend,
      num: int,
  ) -> float:
      """
      Asymptotic P-values for Phillips-Ouliaris t-tests
  
      Parameters
      ----------
      stat : float
          The test statistic
      test_type : {"Za", "Zt", "Pu", "Pz"}
          The test type
      trend : {"n", "c", "ct", "ctt"}
          The trend included in the model
      num : int
          Number of assumed stochastic trends in the model under the null. Must
          be between 2 and 13.
  
      Returns
      -------
      float
          The asymptotic p-value
      """
      test_types = ("Za", "Zt", "Pu", "Pz")
      test_type_key = test_type.capitalize()
      if test_type_key not in test_types:
          raise ValueError(
              f"test_type must be one of: {', '.join(test_types)}"
          )
      trends = ("n", "c", "ct", "ctt")
      if trend not in trends:
          valid = ",".join(trends)
          raise ValueError(f"trend must by one of: {valid}")
      if not 2 <= num <= 13:
          raise ValueError(
              "The number of stochastic trends must be between 2 and 12 (inclusive)"
          )
      key = (test_type_key, trend, num)
      if test_type_key in ("Pu", "Pz"):
          # These are upper tail, so we multiply by -1 to make lower tail
          stat = -1 * stat
      tau_max = PVAL_TAU_MAX[key]
      tau_min = PVAL_TAU_MIN[key]
      tau_star = PVAL_TAU_STAR[key]
      if stat > tau_max:
          return 1.0
      elif stat < tau_min:
          return 0.0
      if stat > tau_star:
          params = np.array(PVAL_LARGE_P[key])
      else:
          params = np.array(PVAL_SMALL_P[key])
      order = params.shape[0]
      x = stat ** np.arange(order)
      return stats.norm().cdf(params @ x)
  
imports: from __future__ import annotations
  from typing import cast
  import numpy
  import pandas
  from scipy import stats
  from statsmodels.iolib.summary import Summary
  from statsmodels.iolib.table import SimpleTable
  from statsmodels.regression.linear_model import RegressionResults
  import arch.covariance.kernel
  from arch.typing import ArrayLike1D, ArrayLike2D, Literal, UnitRootTrend
  from arch.unitroot._shared import KERNEL_ERR, KERNEL_ESTIMATORS, ResidualCointegrationTestResult, _cross_section
  from arch.unitroot.critical_values.phillips_ouliaris import CV_PARAMETERS, CV_TAU_MIN, PVAL_LARGE_P, PVAL_SMALL_P, PVAL_TAU_MAX, PVAL_TAU_MIN, PVAL_TAU_STAR
  from arch.unitroot.unitroot import TREND_DESCRIPTION
  from arch.utility.array import ensure2d
  from arch.utility.io import str_format
  from arch.utility.timeseries import add_trend
label_desc: Use image node_id 5 for calling a global function with example usage: phillips_ouliaris_pval(stat, test_type, trend, num) and returns: stats, int, int